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Derivatives
Basic Properties/Formulas/Rules
d cf x cf x dx
= ′ , c is any constant. ( ) ( )
( f^ x^ g^ x^ ) f^ (^ x^ )^ g^ (^ x)
d (^) n n 1 x nx dx
−
= , n is any number. ( ) 0
d c dx
= , c is any constant.
( f g^ ) f^ g^ f g
′ (^) = ′ + ′ – (Product Rule) 2
f f g f g
g g
( (^ (^ ))) (^ (^ ))^ (^ )
d f g x f g x g x dx
= ′^ ′ (Chain Rule)
( )
( ) (^ )^
d (^) g x g x( ) g x dx
e = ′ e ( ( ))
ln
d g^ x g x dx g x
Common Derivatives
Polynomials
d c dx
d x dx
d cx c dx
d (^) n n 1 x nx dx
−
d (^) n n 1 cx ncx dx
−
Trig Functions
( sin^ ) cos
d x x dx
= ( cos ) sin
d x x dx
2 tan sec
d x x dx
( sec^ ) sec^ tan
d x x x dx
= ( csc ) csc cot
d x x x dx
2 cot csc
d x x dx
Inverse Trig Functions
1 2
sin 1
d x dx (^) x
−
−
1 2
cos 1
d x dx (^) x
− = − −
1 2
tan 1
d x dx x
−
1 2
sec 1
d x dx (^) x x
−
−
1 2
csc 1
d x dx (^) x x
− = − −
1 2
cot 1
d x dx x
− = −
Exponential/Logarithm Functions
( ) ln(^ )
d (^) x x a a a dx
d x x
dx
e = e
ln , 0
d x x dx x
ln , 0
d x x dx x
log , 0 ln
a
d x x dx x a
Hyperbolic Trig Functions
( sinh^ ) cosh
d x x dx
= ( cosh ) sinh
d x x dx
2 tanh sech
d x x dx
( sech^ ) sech^ tanh
d x x x dx
= − ( csch ) csch coth
d x x x dx
2 coth csch
d x x dx
Integrals
Basic Properties/Formulas/Rules
∫ cf^ (^ x dx)^ =c^ ∫ f^ (^ x dx) ,^ c^ is a constant.^ ∫ f^ (^ x^ )^ ±^ g^ (^ x dx)^ =^ ∫ f^ (^ x dx)^ ±∫g^ (^ x dx)
( ) ( ) ( ) ( )
b (^) b
a a ∫ f^ x dx^ =^ F^ x^ =^ F b^ −F^ a where^ F^ (^ x^ )^ =^ ∫f^ (^ x dx)
( ) ( )
b b
a a ∫ cf^ x dx^ =c^ ∫ f^ x dx,^ c^ is a constant.^ (^ )^ (^ )^ (^ )^ (^ )
b b b
a a a ∫ f^ x^ ±^ g^ x dx^ =^ ∫ f^ x dx^ ±∫ g^ x dx
( ) 0
a
a ∫ f^ x dx^ = (^ )^ (^ )
b a
a b ∫ f^ x dx^ = −∫ f^ x dx
( ) ( ) ( )
b c b
a a c ∫ f^ x dx^ =^ ∫ f^ x dx^ +∫ f^ x dx (^ )
b
a ∫ c dx^ =^ c b^ −a
If f (^) ( x (^) ) ≥ 0 on a ≤ x ≤ bthen (^) ( ) 0
b
a ∫ f^ x dx^ ≥
If f (^) ( x (^) ) ≥ g (^) ( x)on a ≤ x ≤ bthen (^) ( ) ( )
b b
a a ∫ f^ x dx^ ≥∫ g^ x dx
Common Integrals
Polynomials
dx = x +c ∫
k dx = k x +c ∫
n n x dx x c n n
= + ≠ −
∫
dx lnx c x
1 x dx lnx c
− = + ∫
n n x dx x c n n
− − + = + ≠ − +
∫
dx lnax b c ax b a
p p p q q q q p q
q x dx x c x c p q
+^ + = + = +
∫
Trig Functions
∫ cos^ u du^ =^ sinu^ +c ∫ sin^ u du^ = −^ cosu^ +c
2 ∫sec^ u du^ =^ tanu^ +c
∫ sec^ u^ tan^ u du^ =^ secu^ +c ∫ csc^ u^ cot^ udu^ = −^ cscu^ +c
2 ∫csc^ u du^ = −^ cotu^ +c
∫ tan^ u du^ =^ ln secu^ +c ∫cot^ u du^ =^ ln sinu^ +c
∫ sec^ u du^ =^ ln sec^ u^ +^ tanu^ +c (^ )
sec sec tan ln sec tan 2
∫^ u du^ =^ u^ u^ +^ u^ +^ u^ +c
∫ csc^ u du^ =^ ln csc^ u^ −^ cotu^ +c (^ )
csc csc cot ln csc cot 2
∫^ u du^ =^ −^ u^ u^ +^ u^ −^ u^ +c
Exponential/Logarithm Functions
u u ∫ e^ du^ =^ e^ +c ln
u u a a du c a
∫ =^ + ∫ln^ u du^ =^ u^ ln(^ u^ )−^ u^ +c
sin ( ) (^2 2) ( sin ( ) cos( ))
au au bu du a bu b bu c a b
∫
e e ( 1 )
u u ∫ u^ e^ du^ =^ u^ −^ e +c
cos ( ) (^2 2) ( cos ( ) sin( ))
au au bu du a bu b bu c a b
∫
e e
ln ln ln
du u c u u
Trig Substitutions
If the integral contains the following root use the given substitution and formula.
2 2 2 2 2 sin and cos 1 sin
a a b x x b
2 2 2 2 2 sec and tan sec 1
a b x a x b
2 2 2 2 2 tan and sec 1 tan
a a b x x b
Partial Fractions
If integrating
( )
( )
P x dx Q x
where the degree (largest exponent) of P x( )is smaller than the
degree of Q x( ) then factor the denominator as completely as possible and find the partial
fraction decomposition of the rational expression. Integrate the partial fraction
decomposition (P.F.D.). For each factor in the denominator we get term(s) in the
decomposition according to the following table.
Factor in Q x( ) Term in P.F.D Factor in Q x( ) Term in P.F.D
ax + b
A
ax + b
( )
k ax + b ( ) ( )
1 2 2
k k
A A A
ax b (^) ax b ax b
2 ax + bx + c 2
Ax B
ax bx c
( )
2 k ax + bx + c ( )
1 1 (^2 )
k k k
A x B A x B
ax bx c (^) ax bx c
Products and (some) Quotients of Trig Functions
sin cos
n m ∫ x^ x dx
1. If n is odd. Strip one sine out and convert the remaining sines to cosines using 2 2 sin x = 1 − cos x, then use the substitution u =cosx 2. If m is odd. Strip one cosine out and convert the remaining cosines to sines
using
2 2 cos x = 1 − sin x, then use the substitution u =sinx
3. If n and m are both odd. Use either 1. or 2. 4. If n and m are both even. Use double angle formula for sine and/or half angle
formulas to reduce the integral into a form that can be integrated.
tan sec
n m ∫ x^ x dx
1. If n is odd. Strip one tangent and one secant out and convert the remaining
tangents to secants using
2 2 tan x = sec x− 1 , then use the substitution u =secx
2. If m is even. Strip two secants out and convert the remaining secants to tangents
using
2 2 sec x = 1 + tan x, then use the substitution u =tanx
3. If n is odd and m is even. Use either 1. or 2. 4. If n is even and m is odd. Each integral will be dealt with differently.
Convert Example : (^) ( ) ( )
6 2 3 2 3 cos x = cos x = 1 −sin x
